1. Field of the Invention
The present invention relates to a carrier synchronization and channel equalization method, and in particular to a joint carrier synchronization and channel equalization method for OFDM systems.
2. The Prior Arts
In the prior art, a device configuration of a baseband equivalent model for Orthogonal Frequency Division Multiplexer (OFDM) is shown in FIG. 1. Wherein, an inverse discrete Fourier transform (IDFT) 10 and a discrete Fourier transform (DFT) 12 of N points are utilized for a baseband multicarrier modulation and demodulation respectively. Each of OFDM symbols is composed of K data symbols Xk,l, wherein, k and l represent the indexes of a sub-channel and a symbol respectively. The subcarrier spacing fΔ is equal to 1/Tu, wherein, Tu is a symbol duration. A guard interval is formed by putting Ng discrete signal samples before each of the transmitted symbols, hereby preventing intersymbol interference between signal symbols. Therefore, the signal sample length of a complete OFDM symbol is Ns=N+Ng, and its corresponding symbol duration is Ts=Tu+Tg. Then, a signal sample in the nth time domain of the lth transmitted OFDM symbol is given by the following formula:
                                                                                          x                                      n                    ,                    l                                                  =                                  x                  ⁡                                      (                                                                                            l                          ⁡                                                      (                                                          N                              +                                                              N                                g                                                                                      )                                                                          ⁢                        T                                            +                                                                        N                          g                                                ⁢                        T                                            +                      nT                                        )                                                                                                                                            =                                                            ∑                                              l                        =                                                  -                          ∞                                                                    ∞                                        ⁢                                          {                                                                        1                          N                                                ⁢                                                                              ∑                                                                                          k                                =                                                                                                      -                                    K                                                                    /                                  2                                                                                                                            k                                ≠                                0                                                                                                                    K                              /                              2                                                                                ⁢                                                                                    X                                                              k                                ,                                l                                                                                      ⁢                                                          ⅇ                                                              j                                ⁢                                                                                                                                  ⁢                                2                                ⁢                                                                                                                                  ⁢                                π                                ⁢                                                                                                                                  ⁢                                                                  nk                                  /                                  N                                                                                                                                                                                        }                                                                      ,                                                    ⁢                                  ⁢                              n            =                          -                              N                g                                              ,          …          ⁢                                          ,                      N            -            1                    ,                                    (        1        )            wherein, T=Tu/N represents a sample interval.
In the structure mentioned above, a channel impulse response of a multi-path fading channel 16 is represented by h(t)=Σihi(t)·δ(t−τi), wherein, hi (t) and τi represent, respectively, an attenuation and a delay spread of the ith path, and then the time domain signal samples xn,l are serially sent into a Digital-to-Analog converter (DAC) 14, and then they are transmitted into channel 16, such that channel noises n(t) exists in channel 16. As such, the channel output can be expressed by y(t)=Σihi (t)·x(t−τi)+n(t), wherein, n(t) is a white Gaussian noise with its expectation value as zero-mean. In this framework, the timing frequency offset between a Digital-to-Analog converter (DAC) 14 and an Analog-to-Digital converter (ADC) 22 is assumed to be ideal. The output signal of the channel is damaged by a carrier frequency offset (CFO) effect, thus upon being sampled by an Analog-to-Digital converter (ADC) 22, the nth reception signal sample of the lth OFDM symbol can be expressed by the following formula:yn,l=y(t)·ej2πΔft|t=l(N+Ng)T+NgT+NT,  (2)formula (2) explains that a carrier frequency offset (CFO) Δf induces a linear increment of phase offsets in the time domain signals.
Subsequently, upon removing the guard interval from the reception signal samples, the remaining reception signal samples are sent into DFT for demodulation processing. Therefore, the kth sub-channel signal of the ith OFDM symbol can be expressed by the following formula:Yk,l=Sk,l+Ik,l+Nk,l  (3)wherein, Sk,l, Ik,l and Nk,l represent a signal term, inter-carrier interference (ICI), and white Gaussian noise on the kth sub-channel respectively. Moreover, Sk,l and Ik,l can be derived as follows:
                                          S                          k              ,              l                                =                                    ⅇ                              j                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                                  ϕ                  kk                                ⁢                                                      N                    -                    1                                    N                                                      ·                          ⅇ                              j2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                                  ϕ                  kk                                ⁢                                                                            IN                      s                                        -                                          N                      g                                                        N                                                      ·                          si              ⁡                              (                                  π                  ⁢                                                                          ⁢                                      ϕ                    kk                                                  )                                      ·                          H                              k                ,                l                                      ·                          X                              k                ,                l                                                    ,                            (        4        )                                                      I                          k              ,              l                                =                                    ∑                                                q                  =                                                            -                      K                                        /                    2                                                                    q                  ≠                  k                                                            K                /                2                                      ⁢                                          ⅇ                                  j                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                      ϕ                    qk                                    ⁢                                                            N                      -                      1                                        N                                                              ·                              ⅇ                                  j2                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                      ϕ                    qq                                    ⁢                                                                                    IN                        s                                            -                                              N                        g                                                              N                                                              ·                              si                ⁡                                  (                                      π                    ⁢                                                                                  ⁢                                          ϕ                      qk                                                        )                                            ·                              H                                  q                  ,                  l                                            ·                              X                                  q                  ,                  l                                                                    ,                            (        5        )            wherein Hk,l is a channel response of the kth sub-channel, and it must satisfy the stationary property in an OFDM symbol. In addition, a local subcarrier frequency offset φqk and an attenuation factor si(πφqk) can be expressed by the following formula:
                              ϕ          qk                =                  q          -          k          +          ɛ                                    (        6.1        )                                          si          ⁡                      (                          π              ⁢                                                          ⁢                              ϕ                qk                                      )                          =                              sin            ⁡                          (                              π                ⁢                                                                  ⁢                ɛ                            )                                            N            ⁢                                                  ⁢                          sin              ⁡                              (                                                      π                    ⁡                                          (                                              q                        -                        k                        +                        ɛ                                            )                                                        N                                )                                                                        (        6.2        )            wherein, ε=ΔfNT is a normalized CFO, and it represents the amount of Δf transferred from the time domain to the frequency domain through DFT. It is evident that the attenuation factor and ICI term are proportional to ε, as shown in formulae (5), (6.1) and (6.2).
In view of the fact that ε in formulae (6.1) and (6.2) is extremely small, while the system enters into a tracking stage, then si(πφkk) is very close to 1 and si(πφqk) almost approaching zero. As such, si(πφkk) in formula (4) can be ignored, and the ICI term in formula (3) can be eliminated. Upon performing the simplification mentioned above, an equivalent channel response {tilde over (H)}k,l on the kth sub-channel can be expressed in polar coordinate as follows:
                                          H            ~                                k            ,            l                          ≈                              ⅇ                          j              ⁢                                                          ⁢              π              ⁢                                                          ⁢                              ϕ                kk                            ⁢                                                N                  -                  1                                N                                              ·                      ⅇ                          j              ⁢                                                          ⁢              2              ⁢              π              ⁢                                                          ⁢                              ϕ                kk                            ⁢                                                                    IN                    s                                    -                                      N                    g                                                  N                                              ·                      G                          H                              k                ,                l                                              ·                      ⅇ                          j              ⁢                                                          ⁢                              θ                                  H                                      k                    ,                    l                                                                                      ≈                              G                          k              ,              l                                ·                      ⅇ                          j              ⁢                                                          ⁢                              θ                                  k                  ,                  l                                                                                        (        7        )            wherein, θHk,l and GHk,l represent the phase and the magnitude distortions on the kth sub-channel Hk,l namely
      H          k      ,      l        =            G              H                  k          ,          l                      ·                  ⅇ                  j          ⁢                                          ⁢                      θ                          H                              k                ,                l                                                        .      In addition, Gk,l=GHk,l and
      θ          k      ,      l        =            π      ⁢                          ⁢              ϕ        kk            ⁢                        N          -          1                n              +          2      ⁢                          ⁢              πθ        kk            ⁢                                    lN            s                    -                      N            g                          N              +                  θ                  H                      k            ,            l                              .      Finally, formula (3) can be rearranged asYk,l={tilde over (H)}k,l·Xk,l+Nk,l  (8)
In order to resolve the adverse effects caused by the CFO and the channel distortion to a received signal, thus a carrier synchronization and a channel equalization techniques are proposed to overcome these problems. In a framework of the prior arts, the carrier synchronization technique is realized by a frequency control loop having an individual frequency detector. However, such a frequency estimation mechanism is not a perfect process. In practice, the carrier frequency jitter will not be zero. Therefore, such a phenomenon will result in the constellation rotation on each sub-channel in an OFDM system, hereby further degrading the system performance. In a practical OFDM system, a carrier phase compensation on each sub-channel is necessary to overcome the constellation rotation.
In addition, in OFDM transmission system, in general, a channel estimation method utilized on each sub-channel is based on the least square (LS) algorithm. However, this method is not very accurate. The residual CFO will destroy the accuracy of the channel estimation on each sub-channel since the residual CFO has not been fully removed. As such, when the carrier synchronization process enters into a tracking stage, the channel information on each sub-channel has to be updated by the least-mean square (LMS) algorithm to track the channel variations.
For the reasons mentioned above, in general, the carrier frequency synchronization and the channel equalization are restrained based on an individual cost function. Furthermore, the mutual interference will occur between the carrier frequency synchronization and the channel equalization to degrade the system performance, namely increase Bit Error Rate (BER) for an OFDM system.
As such, presently, the performance of the carrier frequency synchronization and the channel equalization techniques of the prior art is still not quite satisfactory, and it has much room for improvements.